I was not aware of an LCD "burn-in" mechanism until now:
In the case of LCDs, the mechanics of burn-in are different than plasma and OLED, which develop burn-in from luminance degradation of the light-emitting pixels. For LCDs, burn-in develops in some cases because pixels permanently lose their ability to return to their relaxed state after a continued static usage profile. In more typical usage profiles this image persistence in LCD is only transient.
Both plasma-type and LCD-type displays exhibit a similar phenomenon called transient image persistence, which is sometimes confused with screen burn but is not permanent. In the case of plasma-type displays transient image persistence is caused by charge build-up in the pixel cells (not cumulative luminance degradation as with burn-in), which can be seen sometimes when a bright image that was set against a dark background is replaced by a dark background only; this image retention is usually released once a typical-brightness image is displayed and does not inhibit the display's typical viewing image quality.
The above quote is from WikiPedia, and it makes sense. LCDs use a liquid that contains molecules that align to the electric field, and it could be possible for them to "polymerize" (stick together) when held in alignment with a static display for a long time. From what I read, it is not very common, so perhaps the Kindle Fires had a bad batch of LCDs that are more susceptible to this phenomenon.